# Behaviour of a box of light

Imagine a perfectly reflecting box with a homogenous distribution of photons within it with total energy E (not entirely sure how close to possible this is, but don't worry about that).

I would have thought this box's gravitational field would be very much like the combination of the the box's effect and a mass E/c^2 inside the box. Intuitively, it is quite easy to see the light provides a little extra inertia (because of the effect of motion of the box on the reflections of photons on the walls - the pressure of the photons on the walls is unbalanced if the box is accelerated. Hopefully the arithmetic means that this inertia is the same as there would be from a mass of E/c^2 regardless of the box's dimensions!

In the Einstein field equations, if the box is in an inertial frame I would expect the light within the box to have no direct effect, because the net momentum flux is zero in any direction. But perhaps the reflection of the photons off the walls does all the work - with momentum being continually supplied to the photons from the box at every surface? Is this the non-zero contribution to the Einstein field equations?

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pervect
Staff Emeritus
There's an exact solution in GR for a photon star. This sheds a lot of light on the question of the "box of light", all you need to do is to add an enclosure to this solution and you've got the complete metric of a box of light.

While the general solution of a photon star in GR requires numerical analysis, there's an "easy case" that you can write down the metric of.

https://www.physicsforums.com/showpost.php?p=3578660&postcount=104
https://www.physicsforums.com/showpost.php?p=3580696&postcount=111

The first gives the URL of a paper with the photon star metric.

The second thread discusses how to "enclose" the photon gas star in a shell of exotic matter with zero density and finite tension.

Because the star (and the box of light) is static, you can useful use the notion of the Komar mass to calculate the gravitational field. If you don't have significant time dilation this is just the integral of (rho+3P). The important thing to note here is that pressure causes gravity, and there is a lot of pressure in a box of light that has any significant energy content.

You mentioned momentum flux, but you didn't take into account pressure, which turns out to be important.

Another lesson to be learned - you can define other sorts of mass for this system, with different distributions. However, the Komar mass is the only one that directly relates to the measured gravitational acceleration of a test object.

The Komar mass distribution in the interior of the star is different from the special relativistic mass distribution, and you'll get incorrect results by a factor of about 2:1 if you try to estimate the gravity by E/c^2 and Newton's law.

Outside the shell enclosing the star, E/c^2 provides a good approximation, if you define E properly.